B.8. SPICE BJT β_DC as a Function of Collector Current

SPICE calculates a dc beta, β_DC, when ISE and NE are included in the transistor model. The equation for the calculation uses β_DC = I_C/I_B, where I_B (forward-active mode) includes the ideal and leakage terms, (B.12),

and I_C is the collector current for the forward-active mode, (B.7),

Using (B.7) and (B.12) in the definition of β_DC, we obtain

Equation B.36

which is

Equation B.37

The V_BE variable can be eliminated with

Equation B.38

giving

Equation B.39

where

Equation B.40

and

Equation B.41

Note that for a given transistor, X is just a collection of parameters and the final result is a relatively simple function of I_C if the V_AF effect is neglected such that I_Co ≈ I_C. This is a very good approximation for the project transistors in which V_AF is about 200 V and the maximum |V_BC| is 3 or 4 V. In the project on the investigation of the β_DC dependence on I_C, β_DC is specifically measured for V_BC = 0 (as set automatically by LabVIEW) such that the V_AF effect is eliminated.

The form of (B.39) indicates that β_DC is an increasing function of collector current, I_C. The high-current asymmpote is β_F. The simple SPICE model treats β_DC as a constant β_F, which therefore neglects leakage current. When parameters ISE and NE are included in the transistor model, SPICE uses (B.39) to compute β_DC. Note that the circuit solution of Unit B.6 is substantially more complicated if (B.39) is included to account for the variable nature of β_DC. The project Mathcad file performs this SPICE solution.

Mathcad-generated plots of collector current and the components of base current (mA) are shown in Fig. B.14. I_B1(V_BE) is the component with the same exponent as I_C (ideal transistor base current) and I_B2(V_BE) is the leakage component. That is [(B.12)]

The total I_B(V_BE) is seen to develop at high currents into a plot that is parallel to I_C.

The plot was made with numbers that are representative of the transistor of the project in which the parameters are measured. The diode-connected transistor measurements are made for the range 0.01 < I_C < 1 mA (–2 < logI_C < 1). It is apparent from the plots that the base current for these measurements is made up of a mixture of the two components, or that the base current is in the transition region between leakage-current domination and ideal-current domination.

The pure ideal base current does not develop until I_C > 100 mA. This mode cannot be observed experimentally (even if means to avoid heating are implemented, for example, using pulse measurements). Certain high-level effects, which have not been included in the discussion, would come into play at this level, rendering the theories presented here as marginally valid.

An experimental plot of I_C/I_B = β_DC as a function of I_C is shown in Fig. B.15. A curve fit to the data yields n_E, I_SE, and β_F(104). We note that the maximum β_DC << β_F.